Exercise 10.2 answers: Conic Sections

Open NCERT p. 186Matches NCERT’s answer

1. Compare y² = 12x with the standard form y² = 4ax.
2. So 4a = 12, giving a = 3.
3. A y²-type equation opens along the x-axis, and the coefficient of x is positive, so it opens to the right.
4. Focus = (3, 0); axis is the x-axis, y = 0; directrix is x = −3; latus rectum length = 4a = 12.

AnswerFocus (3, 0); axis y = 0; directrix x = −3; length of latus rectum 12.

Watch this explained “One reading, slowly”, 8:45 into The chord through the focus that measures how open the curve is

Open NCERT p. 186Matches NCERT’s answer

1. Compare x² = 6y with the standard form x² = 4ay.
2. So 4a = 6, giving a = 3/2.
3. An x²-type equation opens along the y-axis, and the coefficient of y is positive, so it opens upward.
4. Focus = (0, 3/2); axis is the y-axis, x = 0; directrix is y = −3/2; latus rectum length = 4a = 6.

AnswerFocus (0, 3/2); axis x = 0; directrix y = −3/2; length of latus rectum 6.

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Open NCERT p. 186Matches NCERT’s answer

1. Compare y² = −8x with the standard form y² = −4ax (a curve opening to the left).
2. So 4a = 8, giving a = 2.
3. A y²-type equation opens along the x-axis, and the coefficient of x is negative, so it opens to the left.
4. Focus = (−2, 0); axis is the x-axis, y = 0; directrix is x = 2; latus rectum length = 4a = 8.

AnswerFocus (−2, 0); axis y = 0; directrix x = 2; length of latus rectum 8.

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Open NCERT p. 186Matches NCERT’s answer

1. Only x² appears (not y²), so the axis of this parabola is the y-axis, that is x = 0.
2. The right side is negative (−16y), so the parabola opens downward.
3. Compare with the standard form x² = −4ay: 4a = 16, so a = 4.
4. For a downward parabola the focus sits on the negative y-axis, at (0, −a) = (0, −4).
5. The directrix is the horizontal line y = a, so y = 4.
6. Length of the latus rectum = 4a = 16.

AnswerFocus (0, −4); axis x = 0; directrix y = 4; length of the latus rectum = 16.

Watch this explained “One reading, slowly”, 8:45 into The chord through the focus that measures how open the curve is

Open NCERT p. 186Matches NCERT’s answer

1. Only y² appears (not x²), so the axis of this parabola is the x-axis, that is y = 0.
2. The right side is positive (10x), so the parabola opens to the right.
3. Compare with the standard form y² = 4ax: 4a = 10, so a = 5/2.
4. The focus sits on the positive x-axis, at (a, 0) = (5/2, 0).
5. The directrix is the vertical line x = −a, so x = −5/2.
6. Length of the latus rectum = 4a = 10.

AnswerFocus (5/2, 0); axis y = 0; directrix x = −5/2; length of the latus rectum = 10.

Watch this explained “One reading, slowly”, 8:45 into The chord through the focus that measures how open the curve is

Open NCERT p. 186Matches NCERT’s answer

1. Only x² appears (not y²), so the axis of this parabola is the y-axis, that is x = 0.
2. The right side is negative (−9y), so the parabola opens downward.
3. Compare with the standard form x² = −4ay: 4a = 9, so a = 9/4.
4. The focus sits on the negative y-axis, at (0, −a) = (0, −9/4).
5. The directrix is the horizontal line y = a, so y = 9/4.
6. Length of the latus rectum = 4a = 9.

AnswerFocus (0, −9/4); axis x = 0; directrix y = 9/4; length of the latus rectum = 9.

Watch this explained “One reading, slowly”, 8:45 into The chord through the focus that measures how open the curve is

Open NCERT p. 187Matches NCERT’s answer

1. The focus (6, 0) lies on the x-axis, so the axis of the parabola is the x-axis.
2. The directrix x = −6 is a vertical line, so the standard form is y² = 4ax.
3. The vertex is midway between the focus and the directrix: midpoint of (6,0) and (−6,0) is (0,0), so a = 6.
4. So 4a = 24, giving the equation y² = 24x.

Answery² = 24x

Watch this explained “Given a condition, find the curve”, 12:50 into Balancing a point against a line, and the four equations that result

Open NCERT p. 187Matches NCERT’s answer

1. The focus (0, −3) lies on the y-axis, so the axis of the parabola is the y-axis.
2. The directrix y = 3 is above the origin while the focus is below it, so the parabola opens downward: x² = −4ay.
3. The vertex is midway between the focus and the directrix: midpoint of (0,−3) and (0,3) is (0,0), so a = 3.
4. So 4a = 12, giving the equation x² = −12y.

Answerx² = −12y

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Open NCERT p. 187Matches NCERT’s answer

1. The vertex is at the origin and the focus (3, 0) lies on the positive x-axis, so the parabola opens to the right, along the x-axis.
2. The distance from the vertex to the focus is a = 3.
3. The standard form for this case is y² = 4ax, so the equation is y² = 12x.

Answery² = 12x

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Open NCERT p. 187Matches NCERT’s answer

1. The vertex is at the origin and the focus (−2, 0) lies on the negative x-axis, so the parabola opens to the left, along the x-axis.
2. The distance from the vertex to the focus is a = 2.
3. The standard form for this case is y² = −4ax, so the equation is y² = −8x.

Answery² = −8x

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Open NCERT p. 187Matches NCERT’s answer

1. The vertex is at the origin and the axis is along the x-axis, so the equation has the form y² = 4ax or y² = −4ax.
2. The point (2, 3) has a positive x-coordinate, so the parabola must open to the right: y² = 4ax.
3. Put x = 2, y = 3 into the equation: 3² = 4a(2), so 4a = 9/2.
4. The equation is y² = (9/2)x.

Answery² = (9/2)x

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Open NCERT p. 187Matches NCERT’s answer

1. The vertex is at the origin and the curve is symmetric about the y-axis, so the equation has the form x² = 4ay or x² = −4ay.
2. The point (5, 2) has a positive y-coordinate, so the parabola must open upward: x² = 4ay.
3. Put x = 5, y = 2 into the equation: 5² = 4a(2), so 4a = 25/2.
4. The equation is x² = (25/2)y.

Answerx² = (25/2)y

Watch this explained “Given a condition, find the curve”, 12:50 into Balancing a point against a line, and the four equations that result