y=34x−37 is the equation of the tangent line at x=4.
The curve is a circle centered at (0,0).
The line y=−43x+6 intersects perpendicularly the tangent line at x=4.
y=34x−325 is the tangent line of the curve at (4,−3).
Explanation
Given the function y=x2−7, then its derivative is
y′=x2−7x
Statement (1) True
The ordinate of the point on the curve with abscissa 4 is f(4)=42−7=3. The gradient of the tangent line at point (4,3)
f′(4)=42−74=34
The equation of the tangent line is found
y−3=34(x−4)
y−3=34x−316
y=34x−316+3
y=34x−37
Statement (2) False
For values of x and y that satisfy the condition, by squaring both sides, then
y=x2−7
y2=(x2−7)2
y2=x2−7
x2−y2=7
The form x2−y2=7 is not the equation of a circle.
Statement (3) True
The line y=−43x+6 has gradient m1=−43. From Statement (1), the gradient of the tangent line at x=4 is m2=34. Because m1m2=−1, both are perpendicular.
Statement (4) False
The curve does not pass through the point (4,−3), because the curve passes through the point (4,3).
