The line 3x + 2y + 1 = 0 meets the hyperbola 4x2-y2 = 4a2 in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are

Question

The line 3x + 2y + 1 = 0 meets the hyperbola 4x2-y2 = 4a2 in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are (A) (-3a2, 8a2) (B) (3a2, 8a2) (C) (-3a2, -8a2) (D) None of these

Tardigrade Admin · Accepted Answer

Let the required point be (x1, y1). The given line 3x + 2y + 1 = 0 ...(1) is chord of contact of the point so it must be same as the line T = 0, i.e. 4xx1 - yy1 = 4a2 ...(2) Comparing the coefficients of (1) and (2), we get (4x1/3) = (-y1/2) = (4a2/-1) ⇒ x1 = -3a2, y1 = 8a2

Q. The line $3 x + 2 y + 1 = 0$ meets the hyperbola $4 x^{2} - y^{2} = 4 a^{2}$ in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are

$(- 3 a^{2}, 8 a^{2})$

$(3 a^{2}, 8 a^{2})$

$(- 3 a^{2}, - 8 a^{2})$

None of these

Q. The line 3x+2y+1=0 meets the hyperbola 4x2−y2=4a2 in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are

(−3a2,8a2)

(3a2,8a2)

(−3a2,−8a2)

None of these

Let the required point be (x1​,y1​). The given line 3x+2y+1=0...(1) is chord of contact of the point so it must be same as the line T=0,i.e.4xx1​−yy1​=4a2...(2) Comparing the coefficients of (1) and (2), we get 34x1​​=2−y1​​=−14a2​⇒x1​=−3a2,y1​=8a2

Q. The line $3 x + 2 y + 1 = 0$ meets the hyperbola $4 x^{2} - y^{2} = 4 a^{2}$ in the points P and Q. The coordinates of the point of intersection of the tangents at P and Q are

$(- 3 a^{2}, 8 a^{2})$

$(3 a^{2}, 8 a^{2})$

$(- 3 a^{2}, - 8 a^{2})$