the curve 12x^2-5y^2=7 intersects the line 2p^2x-5y=7 at the point (1,p). if p=-1, find the coordinates of the other point of intersection.

Graph the two equations, using a slider (DESMOS) for p. See the attached graph. In addition to the two lines, add the point (1,p). Using the slider, adjust p the value of p until the line intersects at point (1,-1). A value of -1 moves the line so that it passes through the first line at (1,-1). p is -1.

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dashataran dashataran · Answer 2 · 2024-02-03T09:00:18+01:00

Answer:

[tex](-3/2,-2)[/tex]

Step-by-step explanation:

[tex]\text{Solution:}\\\text{We have,}\\12x^2-5y^2=7\\\text{or, }3(2x)^2-5y^2=7......(1)\\\\2p^2x-5y=7\\\text{or, }2x-5y=7\\\text{or, }2x=7+5y......(2)[/tex]

[tex]\text{Substituting equation(2) in equation(1),}\\3(7+5y)^2-5y^2=7\\\text{or, }3(49+70y+25y^2)-5y^2=7\\\text{or, }147+210y+75y^2-5y^2=7\\\text{or, }70y^2+210y+140=0\\\text{or, }y^2+3y+2=0\\\text{or, }y^2+2y+y+2=0\\\text{or, }y(y+2)+1(y+2)=0\\\text{or, }(y+1)(y+2)=0\\\text{i.e. }y=-1,-2[/tex]

[tex]\text{Substituting }y=-1\text{ in equation(2),}\\2x=7+5(-1)\\\text{or, }2x=2\\\text{or, }x=1\\\\\text{Substituting }y=-2\text{ in equation(2),}\\2x=7+5(-2)\\\text{or, }2x=7-10=-3\\\text{or, }x=-3/2\\\\\text{So the coordinates of the other point of intersection is (-3/2, -2).}[/tex]